Generic smoothness and irreducibility of Nash CI surfaces

Prove that for generic payoff tables in binary normal-form games, the Nash conditional independence surface—i.e., the Spohn conditional independence variety associated with undirected graphical models that are disjoint unions of cliques and whose partition of players is either (1,…,1,2,2) or (2,2)—is smooth and irreducible.

Background

The paper introduces Nash conditional independence (CI) varieties as Spohn CI varieties for undirected graphical models that are disjoint unions of cliques. In the binary setting, when the partition of players is (1,…,1,2,2) or (2,2), the Nash CI variety is a surface. The authors compute algebro-geometric invariants, such as the degree and canonical bundle, and show that smooth Nash CI surfaces are of general type.

However, Example 4.6 exhibits Nash CI surfaces that are not smooth nor irreducible, suggesting special cases exist. The authors expect this non-generic behavior to be exceptional but note that proving generic smoothness and irreducibility for surfaces is more challenging than for curves (where smoothness and irreducibility were established). They further remark that Bertini’s theorem does not directly resolve this conjecture.